Analyzing genotype-by-environment interaction using curvilinear regression

ABSTRACT: In the context of multi-environment trials, where a series of experiments is con-ducted across different environmental conditions, the analysis of the structure of genotype-by-environment interaction is an important topic. This paper presents a generalization of the joint re-gression analysis for the cases where the response (e.g. yield) is not linear across environments and can be written as a second (or higher) order polynomial or another non-linear function. After identifying the common form regression function for all genotypes, we propose a selection pro-cedure based on the adaptation of two tests: (i) a test for parallelism of regression curves; and (ii) a test of coincidence for those regressions. When the hypothesis of parallelism is rejected, subgroups of genotypes where the responses are parallel (or coincident) should be identified. The use of the Scheffé multiple comparison method for regression coefficients in second-order polynomials allows to group the genotypes in two types of groups: one with upward-facing con-cavity (i.e. potential yield growth), and the other with downward-facing concon-cavity (i.e. the yield approaches saturation). Theoretical results for genotype comparison and genotype selection are illustrated with an example of yield from a non-orthogonal series of experiments with winter rye (Secalecereale L.). We have deleted 10 % of that data at random to show that our meteorology is fully applicable to incomplete data sets, often observed in multi-environment trials.

Keywords: Scheffé multiple comparison method, joint regression analysis, test for parallelism, test of coincidence

Introduction

Farmers and scientists aim to identify superior per-forming genotypes across a wide range of environmental conditions. Here, by environments we mean combina-tions of locacombina-tions and years. The main source of differ-ences between genotypes in their yield stability is the fact that the genotype and environment effects are not additive, i.e. genotype-by-environment interaction (GEI) is present in the data. This interaction can be due to con-trasting drought stress levels, winter low temperature stress, abiotic stresses, growing cycle duration, availabil-ity of nutrients, etc. The GEI can be expressed either as crossovers, when two different genotypes change in rank order of performance when evaluated in different envi-ronments, or inconsistent responses of some genotypes across environments without changes in rank order. The study and understanding of these interactions is a ma-jor challenge for breeders and agronomic researchers attempting to improve complex traits (e.g. yield) across environmental conditions.

Various techniques have been used to analyze the interaction in general and GEI in particular. Read-ers interested in those methods are referred to e.g. Aastveit and Mejza (1992); Annicchiarico (2002); Gauch (1992); Kang and Gauch (1996); Romagosa et al. (2009).

Regression is one of the most popular and most applicable methods used for inference about genotype comparison and selection in the context of multi-envi-ronmental experiments. In regression analysis two sets of variables are used, the first characterizing genotypes, and the second environments. The so-called adjusted

means (or some other genotypic characteristic) for genotypes usually constitute observations of the depen-dent variable. In our illustration we take the original observations of the phenotypic variable as a realization of each dependent variable. The independent variable is defined by environmental indexes, which represent a measurement of productivity. Although Finlay and Wilkinson (1963) defined these environmental indexes as the average over all environments for every geno-type, in this study we compute them with an itera-tive zigzag algorithm (Mexia et al., 1999; Pereira and Mexia, 2010) which leads to the best linear unbiased estimators of the joint regression parameters. In joint regression analysis (JRA), after selecting the variable of interest (e.g. yield), the joint regression model adjusts a linear regression per genotype across all environments (Pereira et al., 2011; Rodrigues et al., 2011), on a syn-thetic variable measuring productivity, the environ-mental index. Several variants of JRA have been pro-posed. The one in which we will be interested here was first proposed by Gusmão (1985), who showed that the precision in analyzing series of randomized block ex-periments was increased by considering environment indexes for individual blocks instead of only one envi-ronmental index per environment. Mexia et al. (1999) introduced the L₂ environmental indexes obtained by minimizing the sum of sums of squares of residuals, both in order to the coefficients of the regressions and to the environmental indexes.

Here a generalization of the joint regression analysis is presented for cases where the response (e.g. the yield) is not linear across environments and can be written as a second (or higher) order polynomial or Received June 30, 2011

Accepted May 04, 2012

1_{Universidade de Évora/Escola de Ciências e Tecnologia}

CIMA-UE – Depto. de Matemática, R. Romão Ramalho, 59 – 7000-671 – Évora – Portugal.

2_{FCT/UNL – Centro de Matemática e Aplicações – 2829-516}

– Caparica – Portugal.

3_{Poznan University of Life Sciences – Dept. of Mathematical}

and Statistical Methods, ul. Wojska Polskiego 28 – 60-637 – Pozna – Poland.

4_{ISLA Campus Lisboa, Laureate International Universities,}

Estrada da Correia, 53 – 1500-210 – Lisboa, Portugal. *Corresponding author <dgsp@uevora.pt>

Edited by: Thomas Kumke

Analyzing genotype-by-environment interaction using curvilinear regression

another non-linear function. In our considerations we shall start with the estimation of the regression func-tions (linear or curvilinear) independently for all geno-types. In the second step the hypothesis of parallelism of regressions for all genotypes is tested. In general it is expected that the hypothesis of parallel regression lines will be rejected because of the presence of GEI in the data. If we reject the parallelism of regression curves, the next step to investigate GEI is to try to find subgroups of genotypes with similar responses to dif-ferent environmental conditions. The genotypes can then be divided into groups based on the Scheffé mul-tiple comparison method for regression coefficients, i.e. the genotypes with similar behavior will be grouped together and the calculations performed for each group separately.

The methodology will be illustrated by using a data set for winter rye (Secalecereale L.) from multi-envi-ronment trials carried out in the years 1997 and 1998 in Słupia Wielka, Poland (52o_{13’ N, 17}o_{13’ E).}

Materials and Methods

The zigzag algorithm

For convenience, let us consider data arranged in a two-way array with I rows and b columns. Suppose

Y_ij is a continuous response variate (e.g. yield) for geno-type i in block j if present. The joint regression mod-el discussed here is an extension of that of Finlay and Wilkinson (1963) where the environmental indexes are computed for each block instead of for each environ-ment. Assuming that the yield vectors are independent, normal, homoscedastic and that genotype i is present in block j (or replicate), the joint regression model can be written as:

Y_ij = a_i+ b_i c_j + e_ij (i=1,..., I; j = 1,..., b) (1)

with a_i the intercept and b_i the slope for genotype i, c_j the block environmental index and e_ij the residuals. These environmental indexes represent the averages over a block/superblock and can be seen as a (spatial) measure of productivity.

Gusmão (1985, 1986) showed that the precision in analyzing series of randomized block experiments was improved considering environmental indexes for individual blocks of only one environmental index per experiment instead of one environmental index per environment. This proposal results in K experiments each with b blocks, i.e. Kb supporting points per regres-sion instead of only K such points used by the classi-cal Finlay-Wilkinson joint regression model (Finlay and Wilkinson, 1963).

To estimate the model parameters, we wish to minimize

, (2)

where p_ij is the weight of genotype i in block j. If the genotype is absent we take p_ij=0 . When the genotype occurs we take

p

ij

=

p

j, j = 1, 2,…, b. These weights may differ from block to block to express differences in the representativeness of the blocks. If there are sev-eral blocks in the same location, their weights will be the same. In the illustration presented in this paper we use 1 and 0 for the weights, because no information was available about the relevance of the importance of blocks.

The zigzag algorithm (Pereira and Mexia, 2010) is used to minimize the loss function (2) iteratively, with respect to a_i , to b_i and to the environmental index

x

_j. For the complete case (i.e. all the genotypes are present in each environment) the average yield per block can be a good initial value for searching the environmental indexes (Gusmão, 1985). When incomplete blocks are used one may take the average yields for the correspond-ing superblock as the initial values. In the worst case any initial values may be taken, since the computation time does not increase much.

We assume that the yield vectors have components normally and independently distributed, so that the zig-zag algorithm will lead to maximum likelihood estima-tors and enable us to make inferences while comparing genotypes.

The zigzag algorithm may be described as follows:

Calculate the initial values for the environ-mental indexes

x

₀b_{, which range within the}

interval [a₀,b₀], wherea0 =Min x{ 01,...,x0b} and

0 { 01,..., 0b} b =Max x x ;

Minimize the function and obtain

and ;

To minimize , minimize the

functions:

, j = 1, 2,…, b,

to obtain the new vector x₀′b _{of new environmental}

indexes;

Standardize the vector of environmental in-dexes to keep the range unchanged. With

{

}

0 01, , 0b

a′=Min x′  x′ , b₀′ =Max x

{

₀₁′, , x₀′b

}

take

( )

0 0

1 0 0 0

0 0

j i

b a

x a x a

b a

− _{′ ′}

= + −

′− ′ ; to obtain the vector x₁b_{, the} new environmental indexes.

Repeat steps (ii) to (iv) until successive sums of sums of squares of weighted residuals differ by less than a fixed value.

At the end of each iteration, a standardization of the adjusted environmental indexes is carried out so that the range does not change from iteration to iteration. The procedure is carried out until the goal function stabilizes. (i)

(ii)

(iii)

(iv)

The environmental indexes adjusted in this way are called L₂ environmental indexes, because the L₂ norm was used. The described zigzag algorithm is a version of the itera-tive algorithms existing in the literature; see for example, Digby (1979); Gabriel and Zamir (1979) and Ng and Wil-liams (2001). Pereira and Mexia (2010) proved the con-vergence of the zigzag algorithm and that the adjusted parameters could be seen as maximum likelihood esti-mators. The alternative algorithms only considered the numerical adjustment.

Test for coincidence

Considering as before Y_ij, the phenotypic observa-tion for genotype i in block j, j = 1,…,b; i = 1,…, I, and

X_j, the environmental index for environment j, can be written as

Y_ij = b_i0 + b_i1z_1j + ... + b_itz_tj + e_ij (3)

where b_{ik (}k = 0, 1,…, t) are the t+1 unknown regression coefficients, z_kj (=f_k(x_j)) are known functions of the envi-ronmental indexes x_j, e_ij are independent and identically distributed random variables following normal distribu-tion with E(e_ij) = 0, for all i, j, and

(4)

After identifying the regression functions for all genotypes, a test of coincidence for regressions can be used to check whether the yield responses for genotypes are similar with respect to environmental indexes. This test can be performed in two stages (Kleinbaum et al., 2008; Williams, 1967): (i) a test for parallelism of regres-sion curves; and (ii) a test of coincidence for those re-gressions.

Equation (3) can be rewritten as

Y_ij = m_i + b_i1(z_1j –z₁) + b_i2(z_2j –z₂) + ... + b_it(z_tj –z_t) +

e_ij, j = 1, ..., b; i = 1, ...I (5)

After centering the observations we have

b_i0= m_i – b_i1z₁ – b_i2 z₂) – ... – b_it z_t. (6)

The null hypothesis to test the parallelism of re-gression functions can be written as

H₀ : b_ik = b_ck, i = 1,…, I; k = 1,…, t, (7)

where b_ck denotes the common kth _{regression coefficient,}

equal for all genotypes.

Let us consider now the case when some of the ob-servations Y_ij are missing. Then, let n_i (≤ b) be the num-ber of environments in which the genotype i is observed, and I i

i

N n

=

=∑ 1

.

Classical regression techniques are used to esti-mate the parameters by the least squares method, inde-pendently for each genotype. Then we have

where b_i is the vector of estimators of regression pa-rameters for the genotype i, Z_i is an (n_i×t) matrix of centered values of explanatory variables, Y_i = [Y_i1, Y_i2, ..., Y_ini]’, i = 1, 2, ..., I. The SS_i,e has n_i – t – 1 degrees of freedom.

Table 1 gives the analysis of variance to test the parallelism of regression functions. The statistic FC I, ,

un-der

H

₀, follows an F central distribution with (I – 1)

t and N – It – I degrees of freedom. After rejecting the hypothesis that all intercepts are equal we can test the hypothesis of the form:

H₀₁ : b₁₀ = b₂₀ = ... b_I0. (8)

To test H01 it is necessary to calculate the estimator

of common regression coefficients b_C (under hypothesis H0). These estimators can be obtained by solving normal

equations of the form

Z’Zb_C = Z’Y (9)

where,

Table 1 – Analysis of variance for parallelism of the regression lines.

Source of variation d. f. Sum of squares Mean square F-ratio

Combined Regression

t

,

SS_{R C} =b Z Y_C′ ′

, ,

SS

MS R C

R C= _t

Between regressions (I – 1)t , ,

1

SS -SS

I

C I i i R C

i=

′ ′

=

∑

b Z Y , ,

SS MS

( 1) C I C I

I t

= −

, ,

MS F

MS C I C I

e =

Combined Residuals N – It – I '

1 1

SS

I I

e i i i i

i= i=

′ ′

=

∑

Y Y −

∑

b Z Y MS SSe e

N It I

=

− −

Total within genotypes N – I , '

1 SS

I

Y I i i

i=

By ' 1 SS

I

C C i i C

i=

′ ′ ′ ′ ′

= Y Y b Z Y− =

∑

Y Y−b Z Y _{we denote the}

comon sum of squares for deviation from regression with νe =N− −I t degrees of freedom.

Let Y _{be the vector of all N observations}

corre-sponding to the vector of t regression parameters and an N× t matrix Z~ _{of observed values of Z. Using the} standard regression technique we obtain the sum of squares for the residualSS E = Y Y ′ −bZ Y ′ .

The overall analysis of variance to test the hypotheses H₀ and H₀₁ is presented in Table 2. The statistic F_I, under H₀₁, follows an F-distribution with I – 1 and N – It –I degrees of freedom.

Plant materials

To illustrate the described methodology, we use the yield data from a winter rye (Secalecereale L.) ex-periment, obtained in multi-environment trials carried out at Słupia Wielka (Poland) in 1997 and 1998. In each design there were four superblocks, with four blocks of four plots each. Each genotype occurred in one plot per superblock. The data set used in this illustration is a sub-set with five genotypes (CHD_296, RAH_596, RAH_697, RAH_797 and URSUS) and 32 blocks. From this two-way table with 32 rows and five columns, 10 % of the data values were deleted to simulate the possibility of having missing values due to pests, animals or other likely factors. The removal of missing values was per-formed so as to produce an identical number of missing cells per genotype. The summary of the two-way data is presented in Table 3.

Results and Discussion

After applying the zigzag algorithm to the non-orthogonal series of experiments described in the previ-ous section, the environmental indexes are obtained and used as independent variable for the regression curves. The response function (i.e. yield) with respect to envi-ronmental index was estimated by several functions and the adjusted coefficient of determination, R2_{, obtained.}

Table 4 shows the adjusted R2 _{for all the considered}

functions and all five genotypes.

Although all the adjusted R2 _{are high and very}

simi-lar, we have decided to use quadratic regression to

ex-press the responses of genotypes because this model is that for all genotypes it gives the best fit of the data, as can be seen from Table 4. The adjusted regression coefficients of the quadratic model, R2 _{and p-values arepresented in}

Table 5. Figure 1 represents the adjusted quadratic regres-sions for the five winter rye genotypes in study.

After rejecting the hypothesis that the regression is not a quadratic curve for each of the genotypes (p < 0.001, Table 5) we are led to test whether the regression curves are parallel for all five genotypes (cf. Table 1). The hypothesis that the regression lines are parallel is rejected (Table 6), as expected after analyzing Figure 1 and from a preliminary analysis where GEI was found in this data.

In the next step of investigating GEI we tried to find subgroups of genotypes where responses are par-allel (or coincident). By simple inspection of the coef-ficients (Table 5), we can consider two groups of geno-types: (1) CHD_296; RAH_797 and URSUS; (2) RAH_596 and RAH_697. In this case this is in accordance with the preliminary analysis presented in Figure 1.

To quantify the differences we can use mul-tiple comparison methods such as Scheffé (Scheffé, 1959; Miller, 1991). When using the Scheffé method, representing by f_{1–a, r, g}the 1 – a quantile of the cen-tral F distribution with r and g degrees of freedom, and 2

, 1,..., , 0,1, 2

im

b

s i= I m= the variance of the re-gression coefficient b_im, the pairs of quadratic re-gression coefficients which satisfy the condition

, m = 0, 1, 2, are different at the significance level a.

Since the regression coefficients b₁ and b₂ do not differ among environments (p < 0.05), we present in Table 7 only the results for the Scheffé multiple com-parison method of the b₀ coefficients.

Table 2 – Overall analysis of variance to test the coincidence of the regression lines.

Source of variation d. f. Sum of squares Mean square F-ratio

Overall regression t SS =b ZY  ′

R

SS

MS R

R= _t

Between intercepts I – 1 SSI =SS - SSE C

SS MS

1

= −I I

I

MS F

MS I I

e =

Between regressions (I – 1)t SS_{C I,} MSC I, FC I,

Residual-combined N – It –I _SS

e MSe

Total within genotypes N – 1 Y Y ′

Table 3 – Descriptive statistics for the five genotypes and the environmental index.

Genotypes ni Mean Std. Dev. Min. value Max. value

CHD_296 29 8.24 2.26 5 12.5

RAH_596 29 9.13 2.80 5.8 13.8

RAH_697 29 9.75 2.71 5.9 14

RAH_797 29 9.62 3.00 5.4 13.6

URSUS 28 10.45 3.57 5.5 15.8

Table 5 – Regression coefficients, coefficient of determination and p-value for the five genotypes.

Genotype Regression coefficients R2 _p-value

b0 b1 b2

CHD_296 -2.049 1.486 -0.037 0.93 <0.001

RAH_596 6.021 -0.354 0.064 0.97 <0.001

RAH_697 6.900 -0.440 0.068 0.97 <0.001

RAH_797 -4.194 1.968 -0.049 0.99 <0.001

URSUS -3.964 1.901 -0.037 0.99 <0.001

Figure 1 – Adjusted quadratic regressions for the five winter rye genotypes in study. The abscissa corresponds to the yield and the ordinate to the environmental index. The dots represent the genotypes and the solid lines the adjusted quadratic regressions.

Table 6 – ANOVA for parallelism of all five quadratic regression lines.

Source of variation d.f. Sum of squares Mean square F-ratio p-value

Combined regression 2 1100.25 550.12

Difference of regressions 8 33.23 4.15 17.74 < 0.001

Combined residuals 129 30.20 0.23

Total within groups 139 1163.67

Genotype

Function CHD_296 RAH_596 RAH_697 RAH_797 URSUS

Linear 0.924 0.958 0.954 0.978 0.983

Logarithmic 0.931 0.928 0.919 0.985 0.985

Inverse 0.919 0.881 0.865 0.972 0.968

Quadratic 0.931 0.972 0.972 0.985 0.986

Compound 0.893 0.966 0.959 0.952 0.950

Power 0.914 0.947 0.935 0.975 0.977

S-Curve 0.918 0.909 0.891 0.981 0.985

Growth 0.893 0.966 0.959 0.952 0.950

Exponential 0.893 0.966 0.959 0.952 0.950

Logistic 0.920 0.942 0.932 0.982 0.907

Table 4 – Adjusted R2

Table 7 – Scheffé multiple comparison tests of the b₀ coefficients.



bi0 – bl0



RAH_596 RAH_697 RAH_797 URSUS

CHD_296 8.07** 8.949** 2.145* 1.915NS

RAH_596 - 0.879NS _{10.215** 9.985**}

RAH_697 - - 11.094** 10.864**

RAH_797 -- - - 0.23NS

*Significant at the 0.05 probability level; **Significant at the 0.001 probability level; NS_{not significant at the 0.05 probability level.}

The groups obtained with the Scheffé method at significance level 1 % are the same as those given by the simple inspection of the coefficients or by analysis of Figure 1.

The multiple comparison method of Scheffé made it possible to divide the genotypes into two groups: one group with upward-facing concavity (i.e. potential yield growth) and other with downward-facing concavity (i.e. the yield approaches saturation). Inspecting the coeffi-cients, especially b₂, it is possible to see the form of the yield curves. If b₂ is positive, the curve will be convex, otherwise concave.

Table 8 shows the common regression coefficients for all genotypes together and each of the two groups obtained using the Scheffé multiple comparison method, while Table 9 gives the ANOVA to test the parallelism of

regression lines for each of the two groups of genotypes. The hypotheses of parallelism between the quadratic re-gressions were rejected for the first group of genotypes (Table 9). However, we do not reject the same hypothesis for the second group. Hence, in this case, we can go one step further and test whether the regressions in the second group are coincident (hypothesis H₀₁). From the ANOVA presented in Table 10 we reject that hypothesis, i.e. al-though the quadratic regression lines are parallel they are distinct. Therefore the adjusted regression functions for the genotypes RAH_596 and RAH_697 can be written as:

2 _ 596

ˆ _{6.229 0.397 0.066} RAH

Y = − x+ x ;

2 _ 697

ˆ

_6.717

_0.397

_0.066

RAH

Y

=

−

x

+

x

,

Table 9 – ANOVA to test the parallelism of regression lines for each of the two groups of genotypes.

Group Source of variation d.f. Sum of squares Meansquare F-ratio p-value

1 Combined Regression 2 699.09 349.54

Difference of Regressions 4 21.13 5.28 22.03 < 0.001

Combined Residuals 77 18.46 0.24

Total within groups 83 738.68

2 Combined Regression 2 413.25 206.62

Difference of Regressions 2 0.008 0.004 0.018NS _0.98

Combined Residuals 52 11.73 0.23

Total within groups 56 424.99

NS_{not significant at the 0.05 probability level.}

Table 10 – ANOVA to test the coincidence of regression functions in the second group (RAH_596 and RAH_697).

Source of variation d.f. Sum of squares Mean square F-ratio p-value

Overall regression 2 415.40 207.70

Between intercepts 1 3.44 3.44 15.26* 0.0003

Between regressions 2 0.008 0.004 0.18NS _0.98

Residual-combined 52 11.73 0.23

Total-within genotypes 57 430.58

*Significant at the 0.001 probability level; NS_{not significant at the 0.05 probability level.}

Table 8 – Common regression coefficients, coefficient of determination and p-values for all genotypes together and each of the two groups (centered data).

Groups of genotypes Regression coefficients R2 _p-value

b0 b1 b2

All 0.000 0.852 0.005 0.95 <0.001

Group 1 0.000 1.628 -0.033 0.95 <0.001

where the b₁ and b₂ are the common regression coeffi-cients from Table 8 and the b₀ coefficients are calculated according to expression (6).

We point out that we did not find groups of geno-types with identical regressions; this may be due to the high level of GEI. All that we can say is that genotypes RAH_596 and RAH_697 have yields parallel, with that of the second genotype a little higher.

Conclusions

The hypothesis of parallelism of regression curves was rejected, which is natural in multi-environment tri-als with interaction between genotype and environment. The main difference in the two subgroups of genotypes where the responses are parallel is that one group had upward-facing concavity (i.e. potential yield growth) and the other had downward-facing concavity (i.e. the yield approaches saturation), which can help breeders in their genotype selection. The approach proposed in this paper is general and applicable to any series of experiments conducted in multi-environment trials or simply to the case of two-way classified data.

Acknowledgments

Dulce G. Pereira is a member of the CIMA-UE, a research center financed by the Science and Technology Foundation, Portugal. The work was partially support-ed by Ministry of Science and Higher Education Grant N N310 447838, and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through PEst-OE/MAT/UI0297/2011 (CMA). We thank the anonymous referees and the As-sociated Editor for their suggestions, which greatly im-proved the paper.

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